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Discussion (49 Comments)Read Original on HackerNews
> Elementary functions typically include arbitrary polynomial roots, and EML terms cannot express them.
If you take a real analysis class, the elementary functions will be defined exactly as the author of the EML paper does.
I've actually just learnt that some consider roots of arbitrary polynomials being part of the elementary functions before, but I'm a physicist and only ever took some undergraduate mathematics classes. Nonetheless, calling these elementary feels a bit of stretch considering that the word literally means basic stuff, something that a beginner will learn first.
- Page 2 and the following example of https://billcookmath.com/courses/math4010-spring2016/math401... (2016)
- Ritt's Integration in Finite Terms: Liouville's Theory of Elementary Methods (1948)
It's not frequent that analysis books will define the class of elementary functions rigorously, but instead refer to examples of them informally.
If a written piece is intended for an audience who knows the jargon, then it's fine to use jargon - in fact it's appropriate and succinct. If it was intended for the laymen, then jargon is inappropriate.
But it seems you're lamenting that this jargon is wrong and that it shouldn't be jargon!?
In a similar vein to this post, the paper points out that general polynomials do not have solutions in E, so of course exp-minus-log is similarly incomplete.
What is intriguing is that we don’t even know whether many simple equations like exp(-x) = x (i.e. the [omega constant]) have solutions in E. We of course suspect they don’t, but this conjecture is not proven: https://en.wikipedia.org/wiki/Schanuel%27s_conjecture
What is a closed-form number?: http://timothychow.net/closedform.pdf omega constant: https://en.wikipedia.org/wiki/Omega_constant
Neither the present article, nor the original one has much mathematical originality, though: Odrzywolek's result is immediately obvious, while this blog post is a rehash of Arnold's proof of the unsolvability of the quintic.
Many things that seem immediately obvious in retrospect weren't obvious before.
The present article could rather have spent time arguing why this isn't like NAND gate functional completeness.
I would have thought the differences lie in the other direction: not that trees of EML and 1 can describe too little, but that they can describe too much already. It's decidable whether two NAND circuits implement the same function, I'm pretty sure it's not decidable if two EML trees describe the same function.
[1] https://en.wikipedia.org/wiki/Richardson%27s_theorem
[1] https://web.williams.edu/Mathematics/lg5/394/ArnoldQuintic.p...
These classes can always safely include all single-valued continuous functions (you cannot even write the _quadratic_ formula in terms of arithmetic and single-valued continuous functions!), but also plenty of non-single-valued functions (e.g. the +-sqrt function which appears in the well-known quadratic formula).
Applying Arnold's proof to the class given by arithmetic and all complex nth root functions (also multivalued) gives the usual Abel-Ruffini theorem. But Arnold's proof applies to the class "all elm-expressible functions" without modification.
But the fact that a single function can represent a large number of other functions isn't that surprising at all.
It's probably obvious to anyone (it wasn't initially to me), but given enough arguments I can represent any arbitrary set of n+1 functions (they don't even have to be functions on the reals - just as long as the domain has a multiplicative zero available) as a sort of "selector":
g(x_0, c_0, x_1, c_1, ... , x_n, c_n) = c_0 * f_0(x_0) + ... + c_n * f_n(x_n)
The trick is to minimize the number of arguments and complexity of the RHS - but that there's a trivial upper-bound (in terms of number of arguments).
And if you want something truly surprising, Riemann's zeta function can approximate any holomorphic function arbitrarily well on the critical strip. So technically you need only _one_ argument.
Can anyone please explain this further? It seems like he’s moving the goalposts.
The post's argument is different than the usual Galois theory result about the unsolvability of the quintic, in that it shows a property that must be true about all EML(x,y)-derived functions, and a hypothetical quintic-solver-function does not have that property, so no function we add to our repertoire via EML will solve it (or any other function, elementary or not, that lacks this property).
You can't solve an equation? Why not just introduce a function that is equal to the solution of the equation! Problem solved.
Can't solve the differential equation x^2 - a = 0? Why not just introduce a function sqrt(a) as its solution! Problem solved.
Can't solve the differential equation y'' = -y? Why not just introduce a function sin(x) as its solution! Problem solved.
A lot of 19th century mathematics was essentially this: discover which equations had solutions in terms of things we already knew about, and if they didn't and it seemed important or interesting enough, make a new name. This is the whole field of so-called "special functions". It's where we also get the elliptic functions, Bessel functions, etc.
The definition of "elementary function" comes exactly from this line in inquiry: define a set of functions we think are nice and algebraically tractable, and answer what we can express with them. The biggest classical question was:
The answer is "no" and Liouville gave us a result which tells us what the answer does look like when the result is elementary.Risch gave us an algorithm to compute the answer, when it exists in elementary form.
I am a professional mathematician, though nowhere near this kind of thing. The result seems amusing enough, but it doesn't really strike me as something that would be surprising. I confess that this thread is the first I've heard of it...
Some of my favorites:
DoctorOetker: "I'm still reading this, but if this checks out, this is one of the most significant discoveries in years."
cryptonektor: "Given this amazing work, an efficient EML operator HW implementation could revolutionize a bunch of things."
zephen: "This is about continuous math, not ones and zeroes. Assuming peer review proves it out, this is outstanding."
[1] https://news.ycombinator.com/item?id=47746610
[2] https://www.reddit.com/r/math/comments/1sk63n5/all_elementar...
I still consider the article important, as it demonstrates techniques to conduct searches, and emphasizes the very early stage of the research (establishes non-uniqueness for example), openly wonders which other binary operators exist and which would have more desirable properties, etc.
Sometimes articles are important not for their immediate result, but for the tools and techniques developed to solve (often artificial or constrained) problems. The history of mathematics is filled with mathematicians studying at-the-time-rather-useless-constructions which centuries or millennia later become profound to human interaction. Think of the "value" of Euclid's greatest common divisor algorithm. What starts out as a curiosity with 0 immediate relevance for society, is now routinely used by everyone who enjoys the world wide web without their government or others MitM'ing a webpage.
If the result was the main claimed importance for the article, there would be more emphasis on it than on the methodology used to find and verify candidates, but the emphasis throughout the article is on the methodology.
It is far from obvious that the tricks used would have converged at all. Before this result, a lot of people would have been skeptical that it is even possible to do search candidates this way. While the gradual early-out tightening in verification could speed up the results, many might have argued that the approach to be used doesn't contain an assurance that the false positive rate wouldn't be excessively high (i.e. many would have said "verifying candidates does not ensure finding a solution, reality may turn out that 99.99999999999999999% of candidates turn out not to pass deeper inspection").
It is certainly noteworthy to publish these results as they establish the machinery for automated search of such operations.
> If this is true, then this blog post debunking EML is going to up-end all of mathematics for the next century.
This is very concerning for mathematics in general.
1: https://news.ycombinator.com/item?id=47775105
It wouldn't be a math discussion without people using at least two wildly different definitions.
I think it really comes down to what set of functions you are calling "elementary".
(I'm not a mathematician, so don't expect me to have an opinion as far as that goes. But the author also writes well in English, and that language we do share.)
> In layman’s terms, I do not consider the “Exp-Minus-Log” function to be the continuous analog of the Boolean NAND gate or the universal quantum CCNOT/CSWAP gates.
But is there actually a combination of NANDs that find the roots of an arbitrary quintic? I always thought the answer was no but admittedly this is above my math level.
Interestingly, the abs (absolute value) function is non-elementary. I wonder if exp-minus-log can represent it.
Admittedly this may be above my math level, but this just seems like a bad definition of elementary functions, given the context.
And for multivariate polynomials, the roots are uncomputable due to MRDP theorem.
Also I'd be glad to see a specific example of a function, considered elementary, which is not representable by EML.
It could be hard, and in any case, thanks for the article. I wish it would be more accessible to me.
f(a,b,c,d,e) = the largest real solution x of the quintic equation x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0
There's not a simple formula for this function (which is the basic point), but certainly it is a function: you feed it five real numbers as input, and it spits out one number as output. The proof that you can't generate this function using the single one given looks like some fairly routine Galois theory.
Whether this function is "considered elementary" depends on who you ask. Most people would not say this is elementary, but the author would like to redefine the term to include it, which would make the theorem not true anymore.
Why any of this would shake the foundations of computer engineering I do not know.
As for why this could be important... we sometimes find new ways of solving old problems, when we formulate them in a different language. I remember how i was surprised to learn how representation of numbers as a tuple (ordered list of numbers), where each element is the remainder for mutually prime dividers - as many dividers as there are elements in the tuple - reduces the size of tables of division operation, and so the hardware which does the operation using thise tables may use significantly less memory. Here we might have some other interesting advantages.
A function which solves a quintic is reasonably ordinary. We can readily compute it to arbitrary precision using any number of methods, just as we can do with square roots or cosines. Not just the quintic, but any polynomial with rational coefficients can be solved. But the solutions can't be expressed with a finite number of draws from a small repertoire of functions like {+, -, *, /}.
So the question is, does admitting a new function into our "repertoire" allow us to express new things? That's what a structure theorem might tell us.
The blog post is exploring this question: Does a repertoire of just the EML function, which has been shown by the original author to be able to express a great variety of functions (like + or cosine or ...) also allow us to express polynomial roots?
At least eml can express the quintic itself, just like the above mentioned operators can
https://en.wikipedia.org/wiki/Template:Mathematical_expressi...
Don't have anything for the perfect numbers though.
AFAIU the original paper is a result in the field of symbolic regression. What definition of elementary function do they use?